The probability of a man hitting a target is $\frac{2}{3}$. What is the minimum number of times he must fire so that the probability of hitting the target at least once is more than $90 \%$?

The mean and variance of a binomial variate $X$ are $16/5$ and $48/25$ respectively. If $P(X > 1) = 1 - K (3/5)^7$, then $5 K =$

In a hurdle race,a player has to cross $10$ hurdles. The probability that he will clear each hurdle is $\frac{5}{6}$. What is the probability that he will knock down fewer than $2$ hurdles?

In a workshop,there are five machines and the probability of any one of them to be out of service on a day is $\frac{1}{4}$. If the probability that at most two machines will be out of service on the same day is $\left(\frac{3}{4}\right)^{3} k$,then $k$ is equal to

$A$ fair coin is tossed $99$ times. If $X$ is the number of times head occurs,then $P[X=r]$ is maximum when $r=$

Let $C_1$ and $C_2$ be two biased coins such that the probabilities of getting a head in a single toss are $\frac{2}{3}$ and $\frac{1}{3}$,respectively. Suppose $\alpha$ is the number of heads that appear when $C_1$ is tossed twice,independently,and $\beta$ is the number of heads that appear when $C_2$ is tossed twice,independently. Then,the probability that the roots of the quadratic polynomial $x^2 - \alpha x + \beta$ are real and equal is: